Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 12 / c

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- Solution c

Solution

For a nonzero level

h

, completing the square gives

(x - \frac{1}{2 h})^{2} + (y - \frac{1}{2 h})^{2} = \frac{1}{2 h ^{2}} - 1.

Thus the level sets of a linear numerator over a radial quadratic are circles centred on

y = x

, real only for

0 < ∣ h ∣ \leq 1/ 2

. The extreme levels shrink to points. The zero contour is the straight line

x + y = 0

, separating positive and negative heights; near it, small nonzero levels are large circles. Heights tend to zero at infinity.

Figure 1.
Circular nonzero contours around the two extrema of a rational height function
. Original contour plot. Blue positive and orange negative circles surround the marked maximum and minimum; the black line is the zero contour. Circle centres lie on $y = x$ . Details.

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